Question

A vector \[\vec{r}\] is inclined at equal acute angles to x-axis, y-axis and z-axis.
If |\[\vec{r}\]| = 6 units, find \[\vec{r}\].

Answer 1

Suppose, vector \[\vec{r}\] makes an angle \[\alpha\] with each of the axis OX, OY and OZ.
Then, its direction cosines are \[l = \cos \alpha , m = \cos \alpha\] and \[n = \cos \alpha\] i.e. \[l = m = n .\]
\[\text{ Now, }l^2 + m^2 + n^2 = 1\]
\[ \Rightarrow l^2 + l^2 + l^2 = 1\]
\[ \Rightarrow 3 l^2 = 1\]
\[ \Rightarrow l^2 = \frac{1}{3}\]
\[ \Rightarrow l = \pm \frac{1}{\sqrt{3}}\]
\[\text{ Since, }\vec{r}\text{ makes acute angle with the axis .} \]
Hence, we take only positive value . 
Therefore,  

\[\vec{r} = \left| \vec{r} \right| \left( l \hat{i} + m \hat{j} + n \hat{k} \right)\]

\[ \vec{r} = 6 \left( \frac{1}{\sqrt{3}} \hat{i} + \frac{1}{\sqrt{3}} \hat{j} + \frac{1}{\sqrt{3}} \hat{k} \right)\]

\[ = 2\sqrt{3} \left( \hat{i} + \hat{j} + \hat{k} \right)\]